1What is the transpose of a matrix? Choose from: a change of rows only, a change of columns only, a change of signs only, a change of both rows and columns.Show Answer

2If \( C = \begin{bmatrix} 5 & -1 & -2 \\ 4 & 6 & 8 \end{bmatrix} \), which matrix is \( C^T \)?Show Answer

3For the symmetric matrix \( A = \begin{bmatrix} 5 & -3 \\ -3 & 4 \end{bmatrix} \), what is \( A^T \)?Show Answer

4If \( P = \begin{bmatrix} 0 & 3 \\ 2 & 1 \\ 4 & 7 \end{bmatrix} \) and \( Q = \begin{bmatrix} 3 & -5 \\ -6 & 1 \\ 8 & 2 \end{bmatrix} \), what is \( (P + Q)^T \)?Show Answer

5Find the transpose of \( R = \begin{bmatrix} 3 & 2 & 1 \end{bmatrix} \).Show Answer

6Find the transpose of \( W = \begin{bmatrix} -3 \\ 4 \\ 5 \end{bmatrix} \).Show Answer

7Find the transpose of \( X = \begin{bmatrix} -1 & 5 \\ 3 & 7 \end{bmatrix} \).Show Answer

8Find the transpose of \( Y = \begin{bmatrix} 3 & 4 & 7 \\ 2 & -1 & 10 \end{bmatrix} \).Show Answer

9Find the transpose of \( Z = \begin{bmatrix} 0 & 1 \\ 5 & -4 \\ 4 & 5 \end{bmatrix} \).Show Answer

10Find the transpose of \( M = \begin{bmatrix} 6 & 7 & 1 \\ 0 & 5 & 2 \\ -1 & 3 & 4 \end{bmatrix} \).Show Answer

11If \( P = \begin{bmatrix} 9 & 4 \\ 8 & 5 \end{bmatrix} \), prove that \( (P^T)^T = P \).Show Answer

12If \( M = \begin{bmatrix} p & q & r \\ a & b & c \\ x & y & z \end{bmatrix} \), find \( M^T \) and \( (M^T)^T \).Show Answer

13For the same matrix M, write the relation between M and \( (M^T)^T \).Show Answer

14If \( A = \begin{bmatrix} 1 & 3 \\ -2 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & -3 \\ 2 & 5 \end{bmatrix} \), find \( A^T + B^T \).Show Answer

15For the same A and B, find \( (A + B)^T \).Show Answer

16For the same A and B, can \( (A + B)^T \) and \( A^T + B^T \) be written as equal? Give a reason.Show Answer

17If \( A = \begin{bmatrix} 1 & 3 \\ 5 & 7 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix} \), prove that \( (A + B)^T = A^T + B^T \).Show Answer

18For the same A and B, can \( A + B \) be called a symmetric matrix? Give a reason.Show Answer

19If \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), \( B = \begin{bmatrix} 4 & -3 \\ -2 & -5 \end{bmatrix} \) and \( A^T = B \), find the values of a, b, c and d.Show Answer

20Find the transpose of the zero matrix \( O = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \). Is \( O^T \) equal to O?Show Answer

21Find \( I^T \) for the unit matrix of order \( 3 \times 3 \) and say what you notice.Show Answer

22The matrix \( \begin{bmatrix} 3 & x + 2 \\ 5 & 1 \end{bmatrix} \) is symmetric. Find x.Show Answer

23A matrix A has order \( 4 \times 2 \). What is the order of \( A^T \), and what is the order of \( (A^T)^T \)?Show Answer

24If \( A = \begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix} \) and \( k = 4 \), verify that \( (kA)^T = kA^T \).Show Answer